Nonequilibrium Steady-State Analysis of a Boundary-Driven Interacting Quantum Spin Chain

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T

ESI DI

L

AUREA

M

AGISTRALE

Nonequilibrium Steady-State

Analysis of a Boundary-Driven

Interacting Quantum Spin Chain

Candidata:

Francesca C

OLLU

Relatore:

Dr. Davide R

OSSINI

Facoltà di Scienze

Corso di Laurea Magistrale in Fisica

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iii

Introduction

The theory of open quantum systems, i.e. physical systems coupled to an en-vironment with which they exchange information, is a backbone of modern research in quantum mechanics. The reason of this lies in the fact that hav-ing a system completely isolated from the environment is an idealization, because every real system is open, thus it does not obey a perfectly unitary quantum dynamics. Indeed, the state of the system is not well represented by a ket vector which evolves obeying the Schrödinger equation, but it is rather described by a density operator, the dynamics of which is determined by the Liouville-Von Neumann master equation.

Studying the dynamics of an open quantum system is not an easy prob-lem, because of the complexity of the interactions between the system and environment. Since the scientific world has been able to perform experiments on nanoscopic scale, where the laws of classical mechanics no longer give a correct description of the underlying physics, the need to overcome this lim-itation and to understand open quantum systems has become mandatory.

Over the years analytical and numerical strategies have been developed in order to tackle this highly non trivial task. In addition to these more “tradi-tional” approaches, over the last two decades the field of quantum simulation has been revealing as a new and intriguing alternative: this paves the way to reproduce Hamiltonian models by means of suitable experimental platforms that can be controlled with a high degree of accuracy. Several proposals have been put in practice, as superconducting circuits with QED cavities or plat-forms that make use of ultracold atoms in optical lattices. These simulators also allow to realize spin systems; this is one of the reasons why studying them is particularly important.

In this thesis we address a boundary-driven Heisenberg chain: a proto-typical interacting quantum model in one dimension. In particular, we have investigated the properties of a XYZ Heisenberg spin-1/2 chain coupled to a pair of dissipators acting only on the edges of the system. In such a sys-tem two kinds of dynamics compete: the Hamiltonian one and the dissi-pative one. Due to the complexity of treating a combination of interacting

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2 Introduction and dissipative problems, following an analytic route is generally unfeasi-ble. Therefore, we have performed a purely numerical analysis. We have studied the steady-state as the asympotic long-time solution of the master equation mentioned above; in this state, we have analyzed three observables under different regimes of dissipation and of Hamiltonian evolution: mag-netization profile, two-point correlation function, spin current.

In order to develop this analysis, three different numerical methods have been employed. The fundamental use of the matrix product operators (MPO) method has allowed us to compute the physical quantities for all lengths of the chain examined in this thesis (8, 12, 16 sites). A preexisting code has been adjusted for the problem under consideration. The quantum trajectories (QT) method has been useful to compare the results obtained from MPO for 8-sites chain and to explore the plausibility of some results for larger chains. The corner-space renormalization (CSR) method has been studied and imple-mented but it seems not to be useful for this kind of quantum systems. In particular, it seems to work more properly when all the spins of the chain are coupled to dissipators. For the last two numerical methods, a code has been written and implemented from scratch, during this thesis work.

This thesis is organized as follows.

In the first chapter we present the basic concepts of the theory of open sys-tems, introducing the notion of pure and mixed states and the density matrix formalism. In particular, we talk about dynamics of open systems, described by a master equation derived under the condition of having a Markovian evolution for the system. In the end of the chapter, technical concepts con-cerning quantum systems are introduced.

In chapter 2, we go through the most recent development in the field of quantum simulator concluding with some examples of experimental plat-forms that simulate spin systems.

Chapter 3 deals with numerical approaches in solving open many-body system problems, with a detailed report about the three numerical methods used in the present work.

Chapter 4 and chapter 5 contain the original part of the work; after a de-scription of the model under study, three observables are analyzed: mag-netization profile, two-point correlation function and spin current. We also investigate the possible presence of a phase transition, when varying suitable Hamiltonian parameters. In the last chapter, there is a report on the perfor-mances of the three methods used, with explanations about the choice of the

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3 parameters in order to obtain converging results.

Finally, in the conclusions we summarize our results and discuss possible perspectives of this work.

The appendix A contains the analysis of the CSR algorithm and its pseu-docode.

The appendix B contains supplemental material, useful to achieve a better understanding of the analysis of magnetization profile, correlations and spin transport of the system under consideration.

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5

Chapter 1

Open Quantum Systems

The motivation for studying open systems is that all realistic systems are open.

John Preskill

The world we live in is constituted by parts that communicate with each other and exchange information. The difficulty of describing it, lies in the complexity of the interactions between those parts. To overcome this prob-lem, simplifications and easier mathematical representations have been de-veloped: in the following we will see some of them.

This chapter has the purpose of giving a recap of the most useful char-acteristics of open quantum systems, in the perspective of studying an open many-body system and, as stated before, a 1D spin chain, in particular. First of all, in section 1.1 we will specify the difference between closed and open systems and we will derive the density matrix formalism; then, in section 1.2 we will introduce the master equation useful for describing the dynamics of open quantum systems; finally, in section 1.3 we will see some impor-tant technical concepts that we will use effectively in the original part of the present work.

1.1 Mathematical Background

Before we explore the field of open quantum systems, let us assess the basic differences between a closed and an open quantum system and why different approaches are required to treat these two distinct kinds of systems. First of all, let us define a closed system as a physical system which does not exchange any information with its surroundings. On the contrary, an open system is a physical system that interacts with the environment in which it is embedded.

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6 Chapter 1. Open Quantum Systems In this section, we will briefly look at the different approaches in the study of the dynamics in closed and open quantum systems.

1.1.1 Pure and Mixed States: the Density Matrix Formalism

In addition to the normally used formalism for handling pure states, i.e. states that can be described by a single wave function, it is useful to intro-duce the density matrix formalism for managing the so-called mixed states, although - as we will see - it can be used also for pure states [1].

Quantum systems prepared in such a way that their state vector being obtained performing a maximal measurement, in the sense that the maximum possible information has been acquired (all values of a complete set of com-muting observables have been ascertained), are in a pure state.

Quantum systems for which the maximum possible information is not available, are said to be in mixed state and are called statistical mixtures.

Let us consider a system consisting of an ensemble of N possible realiza-tions α = 1, 2, . . . , N. We suppose that every realization of the system is in a pure state|ψαi. Then, we choose a complete set of basis vectors|ni, such that

∑n|ni hn| = 1. Let us expand the pure states in the basis{|ni}:

|ψαi =

∑

n

c(nα)|ni,

where the coefficients c(nα) are such that

∑

n

|c(nα)|2 =1.

Now let us consider an observable represented by an operator A; its expec-tation value in the pure state|ψαiis:

hAi_α = hψα|A|ψαi (1.1) =

∑

n

∑

n0 c(nα)c (α)∗ n0 hn0|A|ni (1.2) =

∑

n

∑

n0 hn|ψαi hψα|n 0_{i h} n0|A|ni. (1.3) The average value of A over the ensemble is

hAi =

N

∑

α=1

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1.1. Mathematical Background 7 where the coefficients wα are the statistical weight of the pure states|ψαi, i.e.

the probability of finding the system in this state. So the wα are such that

0≤wα ≤1 (1.5) and N

∑

α=1 wα =1.

Using the result 1.2 in the 1.4, we obtain:

hAi = N

∑

α=1

∑

n

∑

n0 wαc (α)∗ n0 c (α) n hn0|A|ni (1.6) = N

∑

α=1

∑

n

∑

n0 hn|ψαiwαhψα|n 0_{i h} n0|A|ni. (1.7) We now introduce the density operator

ρ=

N

∑

α=1

|ψαiwαhψα|, (1.8)

whose representation in the basis{n}gives us the density matrix: ρnn0 = hn|_ρ|n0i = N

∑

α=1 wαc (α)∗ n0 c (α) n . (1.9)

So, returning to the 1.7 in the definition of the density matrix, we observe that:

hAi =

∑

n

∑

n0

hn|ρ|n0i hn0|A|ni = Tr(ρA).

It is worth stressing the fact that the knowledge of the density matrix enables us to obtain the ensemble average of A.

Anyway, this leads us to the first of the three fundamental characteristics of the density matrix, that is:

Tr(ρ) =1, (1.10)

that is true if we make the assumption that the|ψαiare normalized to unity.

If they are not, then the ensemble average of A is given by

hAi = Tr(ρA)

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8 Chapter 1. Open Quantum Systems The second feature of the density matrix is that it is Hermitian, namely:

hn|ρ|n0i = hn0|ρ|ni∗. (1.12) The third feature arises from the observation of the diagonal elements of ρ:

ρnn = hn|ρ|ni = N

∑

α=1 wα|c (α) n |2, (1.13)

which tells us, with the 1.5, that

ρnn ≥0, (1.14)

i.e. ρ is a positive semi-definite matrix. It is worth stressing the physical interpretation of the diagonal elements; wα is the probability of finding the

system in the pure state |ψαi, while|c

(α)

n |2 is the probability of finding|ψαi

in the state |ni. So, ρnn gives the probability of finding a member of the

ensemble in the state|ni.

1.1.2 Closed Quantum Systems

Quantum mechanics establishes that the time evolution of a state vector|ψ(t)i is predicted by the Schrödinger equation:

i d

dt |ψ(t)i = H(t) |ψ(t)i, (1.15) where H(t)is the Hamiltonian of the closed system; from now on, the Planck’s constant ¯h is set equal to 1. The solution of this equation can be written in terms of the unitary time-evolution operator U(t, t0):

|ψ(t)i =U(t, t0) |ψ(t0)i, (1.16)

where t0<t. If we substitute the equation 1.16 into the 1.15, we obtain:

id

dtU(t, t0) = H(t)U(t, t0), (1.17) subject to the initial condition

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1.1. Mathematical Background 9 We can write the dynamics of a closed system using the density matrix for-malism seen in the previous section. In order to do this, let us assume that at an initial time t0the state of the system is characterized by the density matrix

ρ(t0) = N

∑

α=1

wα|ψα(t0)i hψα(t0)|, (1.19)

that evolves in this way:

ρ(t) = N

∑

α=1 wαU(t, t0) |ψα(t0)i hψα(t0)|U(t, t0) † , (1.20) that is ρ(t) =U(t, t0)ρ(t0)U(t, t0)†. (1.21)

Differentiating with respect to time the last equation we have: d

dtρ(t) = −i[H(t), ρ(t)], (1.22) where [,] indicates the commutator. This is the equation of motion for the density matrix, often called the Liouville-von Neumann equation [2].

1.1.3 Open Quantum Systems

Now we can go into a more specific definition of open quantum systems. An open quantum system can be defined as a system S coupled with another quantum system E, called the environment. Usually, the total system S+E (typically referred to as the universe) is treated as a closed system, so it fol-lows Hamiltonian dynamics. The interactions between the two sub-systems induce an evolution in the state of the open system S, which can no longer be represented by unitary, Hamiltonian dynamics. Following the definitions given in [2], an environment with an infinite number of degrees of freedom such that the frequencies of the modes form a continuum, is called reservoir. Also, a reservoir in a thermal equilibrium state is called bath.

Since S and E are in thermal contact, energy can be exchanged between them; however the interaction is supposed to be weak, so that both S and E at any time can be considered to be in definite energy eigenstates with energies ES and EE respectively. Naming E one of the eigenenergies of the combined

system, for the conservation of energy we have: E =ES+EE.

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10 Chapter 1. Open Quantum Systems

FIGURE1.1: Schematic representation of an open quantum system.

Let dΓ(E)be the total number of distinct states of the combined system with energies in the range[E, E+dE]. So, the probability of finding the combined system with an energy in this range will be1:

dP =CdΓ(E), E0 ≤E ≤E0+∆ (1.23)

and zero otherwise; C is a constant.

If we call dΓS and dΓE the number of states for the system and the

envi-ronment, respectively, we have:

dΓ=dΓSdΓE, (1.24)

so that:

dP=CdΓS(ES)dΓE(EE)δ(E−ES−EE), (1.25)

where the delta function expresses the conservation of energy.

The probability dPn that the combined system is in a state such that the

system S has an energy ES, without regard to the state of the reservoir, can be

obtained integrating over the states of the reservoir: dPn =CdΓS(En)

Z

δ(E−En−EE)dΓE(EE)

=CdΓS(En)∆ΓE(E−En),

(1.26)

where∆ΓE(E−En)is the total number of states of the environment, that can

be written in terms of the entropy SE of the reservoir:

∆ΓE =exp(SE/k),

where k denotes the Boltzmann constant.

1_{This is true because of the postulate of equal a priori probabilities, since the combined}

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1.1. Mathematical Background 11 Since En E, we can write:

SE(EE) = SE(E−En) ' SE(E) −

∂SE(E)

∂E En =SE(E) − En

T , where the temperature T has been defined as:

1

T =

∂S ∂E. Substituting in eq. 1.26, we find that

dPn = A exp(−En/kT)dΓS(En), (1.27)

where A is defined as

A =C exp(SE(E)/k). (1.28)

The probability 1.27 can be written in terms of the density matrix, in this way:

ρnm = A exp(−En/kT)δnm.

Using the normalization condition Tr(ρ) =1, we can find A−1 ≡QS:

QS(T) =

∑

n exp(−En/kT) = Tr[exp(−βHS)], where β= 1 kT

HS is the Hamiltonian of the system S and QS is the partition function.

So, we can write the density operator in the canonical ensemble as:

ρ= 1

QS(T)

exp(−βHS). (1.29)

The average value of an observable O over the ensemble is

hOi =Tr(ρO) = 1 QS Tr

[exp(−βHS)O] = Tr

[exp(−βHS)O]

Tr[exp(−βHS)] . (1.30)

Let us callHS and HE the Hilbert spaces of the system and the

environ-ment, respectively. The Hilbert space of the combined system is given by the tensor productH_S+E = HS ⊗ HE. So, the total Hamiltonian can be written

in this way:

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12 Chapter 1. Open Quantum Systems where HS and HE are the free Hamiltonians of the system and the

environ-ment, respectively, and HI(t) is the Hamiltonian describing the interactions

between S and E.

At this point, we must find a relation between the density matrix ρ ∈ HS+E and ρS ∈ HS; it is given by the partial trace over the environment E2:

ρS =TrE(ρ). (1.32)

This helps us in the study of the dynamics of the so-called reduced density matrix ρS. Indeed, we can write:

ρS(t) =TrE[ρ(t)]; (1.33) observing that we assumed the total system to be closed, it evolves unitarily so we have:

ρS(t) = TrE{U(t, t0)ρ(t0)U(t, t0)†}, (1.34)

where U(t, t0) is the time-evolution operator of the combined system S+

E. So we can use the equation of motion 1.22 and trace over the degrees of freedom of the environment, in order to obtain the equation of motion for the ρS(t):

d

dtρS(t) = −i TrE[H(t), ρ(t)]. (1.35)

1.2 Approximate Description of

Open Quantum Systems

If one considers a many-particles system, where the Hilbert space becomes “large”, solving exactly the equation 1.35 is rather difficult, because of the fact that the environment may represent a reservoir consisting of infinitely many degrees of freedom, or its modes may be neither known nor controllable, for example. In the following, we will derive a simpler and approximate description of the dynamics of an open quantum system.

1.2.1 Quantum Dynamical Semigroups

For reasons that will be clear later, let us rewrite the equation 1.35 as a dy-namical map acting onH_S that connect the states of the system S from time t0

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1.2. Approximate Description of Open Quantum Systems 13 to time t1, that is:

ε(t1,t0) : ρS(t0) 7→ρS(t1). (1.36)

In order to obtain the way in which this map acts on the states, we consider a system consisting in two sub-systems S+E, that are uncorrelated3 at the initial time t=0, so that

ρ(0) = ρS(0) ⊗ρE(0), (1.37)

where ρS(0)is the initial state of the system S and ρE(0)represents the state

of the environment at time t = 0. The evolution of the system S from the time t=0 to the time t>0 is governed by the map, in this way:

ρS(0) 7→ρS(t) = ε(t)ρS(0) ≡TrE{U(t, 0)[ρS(0) ⊗ρE(0)]U†(t, 0)}. (1.38)

Using the spectral decomposition of the density matrix ρEof the environment

ρE =

∑

i

λi|ψii hψi|,

we can perform the partial trace of 1.38 in this basis, obtaining: ρS(t) =

∑

ij λihψj|U(t, 0) |ψiiρS(0) hψi|U†(t, 0) |ψji =

∑

a MaρS(0)Ma†. (1.39)

So, the dynamical map can be expressed in terms of the following so-called operator-sum representation:

ε(ρ) =

∑

a

MaρM†_a, (1.40)

where the{Ma}obey the completeness relation

∑

a

M†_aMa =1 (1.41)

and are called Kraus operators and are defined as [3] Ma =

p

λihψj|U(t, 0) |ψii_a. (1.42) 3_{Initial correlations between environment and the open system play significant role in}

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14 Chapter 1. Open Quantum Systems This object has some essential features that give it the name completely positive trace-preserving map or CPTP map; as its name suggests, its properties4are:

• linearity: ε(αρ1+βρ2) = αε(ρ1) +βε(ρ2); (1.43) • preserves Hermiticity: ρ=ρ† =⇒ ε(ρ) = ε†(ρ); (1.44) • preserves positivity5: ρ ≥0 =⇒ ε(ρ) ≥0; (1.45) • preserves trace: Tr[ε(ρ)] =Tr(ρ). (1.46) The CPTP maps are called by several names; one of them is super-operator, due to its function of connecting operators with operators, rather than vec-tors with vecvec-tors. Another name used in the field of Quantum Information is quantum channel, that originates from communication theory and gives the image of the "channel" throughout the state ρ travels to reach its destination as ε(ρ).

There is a reason why we can say that the CPTP maps form a dynamical semigroup: that is because the associative property is satisfied. Indeed, if ε1

is a CPTP map with N Kraus operators{Aa}and ε2is a CPTP map with M

Kraus operators {Bb}, than ε1◦ε2 has an operator-sum representation with

N M Kraus operators{AaBb}.

1.2.2 The Markovian Evolution

With the final purpose to cast the evolution of the density operator in terms of a differential equation, let us analyze if the evolution of an open quantum system can be described as a Markovian process6, i.e. local in time. In this

4_{A detailed proof of the properties can be found in [5].}

5_{In fact, the map satisfies a stronger property, i.e. it is completely positive; this means}

that if ε has an operator-sum representation with Kraus operators{Ma}, then ε⊗1 has an

operator-sum representation with Kraus operators{Ma⊗1}.

6_{Let us recall what a Markovian process is: a stochastic process is a Markovian process}

if the probability that the random variable X takes the value xn at an arbitrary time tn, is

conditioned only by the value xn−1 that it took at time tn−1 and does not depend on the

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1.2. Approximate Description of Open Quantum Systems 15 context, with Markovian evolution, we denote the evolution of a system S that does not keep track of the state of the environment E; it is a “memoryless” evolution.

The reasons of this lie in the fact that if we want that a (first order) dif-ferential equation in t governs the evolution of ρ(t), ρ(t+dt) must be com-pletely determined by ρ(t).

So, a problem emerges: an open system is dissipative, in the sense that in-formation can flow from the system into the environment but also can flow back from the environment into the system, causing non-Markovian fluctu-ations. This fact seems to prevent the possibility of applying the Markovian model. A possible solution is neglecting the typical correlation time of the fluctuations with respect to the time scale of the evolution that we want to study.

1.2.3 The Lindblad Master Equation

Once we take for granted that the dynamics is Markovian, we can write the evolution for the infinitesimal time interval dt:

ρ(t+dt) = εdt[ρ(t)]. (1.47) Expanding εdtto first order, we get:

εdt=1+dtL, (1.48)

and we find

˙ρ= L[ρ], (1.49)

where the linear map generating time evolutionLis called Liouvillian or Lind-bladian. The quantum channel 1.48 has the operator-sum representation

εdt[ρ(t)] =

∑

a

Maρ(t)Ma†=ρ(t) + O(dt). (1.50)

So we may assume that M0 =1+ O(dt)and the Mais of order

√

dt for a >0. In particular, we can write:

M0=1+ (−iH+K)dt (1.51)

Ma =

√

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16 Chapter 1. Open Quantum Systems where H, K are both hermitian, La, H, K are zeroth order in dt and N is

the dimension of the Hilbert space. Now, we can determine K using the completeness relation 1.41: 1= N2−1

∑

a=0 M_a†Ma =M0†M0+ N2−1

∑

a=1 L†_aLadt =₁+2Kdt+ N2₋₁

∑

a=1 L†_aLadt+ O(dt2), (1.53)

so, keeping terms up to linear order in dt we have:

K = 1 2 N2₋₁

∑

a=1 L†_aLa.

Substituting in equation 1.50, we obtain the Lindblad master equation7[2, 5]:

˙ρ= L[ρ] ≡ −i[H, ρ] − N2−1

∑

a=1 γa 1 2L † aLaρ+1 2ρL † aLa−LaρL†_a . (1.54)

This is the equation that governs the evolution of a quantum system, with the assumption that the evolution is a CPTP map. The quantities γaare

non-negative numbers that have the dimension of an inverse time and have the physical meaning of dissipation rate.

It is clear that the first term of equation 1.54 represents the unitary part of the dynamics generated by the Hamiltonian; the second part, instead, de-scribes the possible evolution that the system may undergo due to the inter-action with the environment. Indeed, the Laare called quantum jump operators

or Lindblad operators.

1.3 Quantifying Correlations in Quantum Systems

In this section, we introduce a few concepts, useful to better understand quantum systems, as we will see in the following chapters.

7_{The derivation of the master equation developed is not formally rigorous; for a more}

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1.3. Quantifying Correlations in Quantum Systems 17

1.3.1 Bipartite Pure States

Schmidt Decomposition and Entanglement

A very useful tool to manipulate pure quantum states is the Schmidt decom-position, that we present here following the line of reasoning of [3].

Theorem 1.3.1. Let us suppose that|ψiis a pure state of a composite system AB, that lives in the Hilbert space given by the tensor product HA⊗ HB. Then there

exist orthonormal bases, the Schmidt bases, {χ(_iA)} and {χ(_jB)}, in HA and HB,

respectevely, such that

|ψi =

∑

i

αi|χ(_iA)i ⊗ |χ_i(B)i. (1.55) The αi are complex numbers called Schmidt coefficients and the number of non-zero αiis called Schmidt number.

Proof. Let|χjiand|χkibe orthonormal bases for systems A and B, respectevely.

So,|ψican be written as|ψi = ∑jkajk|χji |χki. Using the singular value

de-composition, we can write the matrix a constituted by the elements ajk, in

this way: a= udv, where d is a diagonal matrix with non-negative elements called singular values of the matrix, u and v are unitary matrices. So we have

A state |ψi ∈ HA⊗ HB is said to be entagled if it cannot be written as a

tensor product|χ(A)i ⊗ |χ(B)iof states of the sub-systems, i.e. if the Schmidt number is greater than 1. Moreover, if the absolute values of all non-vanishing Schmidt coefficients are equal to each other, the state|ψi is said to be maxi-mally entangled. Conversely, if a state|ψican be written as a tensor product is named product state and this is true if and only if the Schmidt number is equal to 1. We can say that the Schmidt number is a measure of the entanglement between the system A and the system B.

The entanglement is one of the most peculiar manifestations of quantum mechanics. Given a composite quantum system, entanglement quantifies non-classical correlations between measurements performed on well-separa-ted particles. Indeed, while after two classical systems have interacwell-separa-ted, they can be described by two defined states, after two quantum systems have interacted, they can no longer be described independently of each other. This kind of systems violate the Bell’s inequality, which can be explained in a heuristic way by means of a thought experiment [3].

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18 Chapter 1. Open Quantum Systems Charlie prepares in some way two particles, then he sends one particle to Alice and the second particle to Bob. Once Alice receives her particle, she performs a measurement on it. She has at her disposal two different measurement apparatuses, so she could choose to do one of two different measurements on physical properties, that we call PQ and PR. She chooses

randomly what measurement to do first. We suppose for simplicity that the measurements can each have one of two outcomes, +1 or −1. Let us call Q and R the outcomes of the measurements. Bob does the same thing, but he will perform the measurements on the properties PS and PT, having as

outcomes S and T, respectively. Again, we suppose for simplicity that the measurements can each have one of two outcomes, +1 or −1. The timing of the experiment is arranged so that Alice and Bob do their measurements at the same time. Therefore, the measurement which Alice performs cannot disturb the result of Bob’s measurement (or vice versa). Doing some simple algebra [3], we obtain the Bell’s inequality:

hQSi + hRSi + hRTi − hQTi ≤2,

where hi denotes the mean value of a quantity. Now, Charlie prepares a quantum system of two qubits in the state

|ψi = |00i + |√ 11i

2 ,

and then he passes a qubit to Alice and a qubit to Bob. Now, if Alice per-formes a measurement obtaining |0i or |1i, Bob will get the same measure obtained by Alice, since|00iis the only state in which the qubit measured by Alice is in the state|0i. If they perform measurements of some observables Q, R, S, T, they find out that the Bell’s inequality is violated. This is an exam-ple of maximally entangled quantum states of two qubits, called in this way because it violates maximally the inequality written above. Since about the 80s, this kind of experiments has been performed in laboratory [6], proving the quantum mechanical predictions.

The notion of entanglement is useful also for the characterization of the efficiency of some numerical methods for the simulation of quantum many-body systems, in particular the ones based on tensor networks, as the matrix product operators method, which we will use in the following.

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1.3. Quantifying Correlations in Quantum Systems 19

Von Neumann Entropy

A crucial tool useful to measure the entanglement of a bipartite pure state is the quantum entropy, defined by Von Neumann entropy in this way:

S(ρ) ≡ −Tr(ρlog₂ρ), (1.56) where ρ coincides with the reduced density matrix ρA of a subsystem A

which can be written as ρA = TrB(|ψi hψ|). If λi are the eigenvalues of ρ,

we can write:

S(ρ) = −

∑

i

λilog₂λi. (1.57)

Let us highlight some important properties of the von Neumann entropy [3, 2]:

• S(ρ) ≥ 0, where the equality sign holds if and only if ρ is a pure state; • in a D-dimensional Hilbert space, the entropy is S(ρ) ≤ log2D, where

the equality sign holds if and only if the system in the completely mixed state ρ=1/D;

• if a composite system AB is in a pure state, S(A) =S(B);

• it is a concave functional ρ 7→ S(ρ) on the space of density matri-ces. This means that for any collection of ρi and numbers λi satisfying

∑iλi = 1, one has S(∑iλiρi) ≥ ∑iλiS(ρi), where the equal sign holds

if and only if all ρi are equal to each other. This property means that

the uncertainty of the state ρ=∑_iλiρiis greater or equal to the average

uncertainty of the states ρithat constitute the total mixture;

• if ρ associated to Hilbert space H = HA⊗HB is the state of a system

and ρA =TrBρand ρB =TrAρare the states of its sub-systems, the von Neumann entropy obeys the so-called subadditivity condition, i.e.

S(ρ) ≤ S(ρA) +S(ρB),

where the equality sign holds if and only if ρ describes an uncorrelated state. This means that the uncertainty about the product state is greater than that of the total system; this is because the operation of trace over the sub-systems causes a loss of information on correlations between the sub-systems, therefore the increase of entropy.

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20 Chapter 1. Open Quantum Systems

1.3.2 Bipartite Mixed States

Subsystems of a many-body pure state will generally be in a mixed state ρ. In the case of mixed states, the measure of entanglement is unclear and diverse; four important representatives [7, 8] are the following:

entanglement cost, an asymptotic limit of the number k of maximally entan-gled states used to extract n copies of ρ, minimized over all the possible local quantum operations with classical communication (LOCC) ma-nipulations:

EC(ρ) = inf LOCCnlim→∞

k

n; (1.58)

distillable entanglement, an asymptotic limit of the number k of maximally entangled states used to distill n copies of ρ, minimized over all the possible LOCC manipulations:

ED(ρ) = inf LOCCnlim→∞

k

n; (1.59)

entanglement of formation, average of the entanglement of pure states which form the mixture, minimized over all the possible decompositions:

EF(ρ) =min

∑

j

pjS(ρA,j), (1.60)

where the minimum is taken over all the realizations of the state

ρ=

∑

j

pj|ψji hψj| (1.61)

and S(ρA,j)is the Von Neumann entropy of the reduced density matrix ρA,j=TrB(|ψji hψj|);

relative entropy of entanglement, distance of ρ from the nearest mixed sep-arable state σ:

ER(ρ) = min

σ∈sep.Tr

[ρ(log₂ρ−log₂σ)]. (1.62) Moreover, the Von Neumann entropy can be used also for bipartite mixed states; that is, we can calculate

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1.3. Quantifying Correlations in Quantum Systems 21 where ρ is the reduced density matrix of a subsystem that we can express in this way: ρ=TrB( N

∑

j=1 |ψji hψj|). (1.64)

In this case it is called operator space entanglement entropy and it does not di-rectly measure the entanglement but it is a good quantifier of the functional-ity of matrix product operators method, as we will see in section 3.3.

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23

Chapter 2

Quantum Simulators of Open

Many-Body Systems

Can a quantum system be probabilistically simulated by a classical universal computer? [. . . ] The answer is certainly, no!

Richard P. Feynman

Open quantum many-body systems represent a challenging area of study, due to the exponential growth of the number of freedom degrees with respect to the number of the constituent particles. The scientific community has built many strategies to overcome this obstacle, by developing and refining several analytical and numerical methods. In addition to these, another approach was proposed, based on the use of quantum simulators [9].

In the present chapter we report a review of some of the most relevant ideas in the field of quantum simulators; in particular, the last section focuses on the simulation of spin systems, of remarkable interest in view of the model studied in this work.

2.1 Quantum Many-Body Systems

Quantum many-body systems are systems made up by a certain number of interacting particles. Typical quantum many-body systems are represented by Hubbard-like models or spin chains. A prototype of these systems is de-scribed by the non-dissipative Bose-Hubbard model [10]: an interacting bo-son gas in a lattice potential. The Hamiltonian reads:

H = −J

∑

hi,ji a†_iaj+

∑

i eini+ 1 2U

∑

_i ni(ni−1), (2.1)

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24 Chapter 2. Quantum Simulators of Open Many-Body Systems where the notationhi, jidenotes the sum over nearest-neighbour sites; aiand

a†_i are the bosonic creation and annihilation operators respectively, on the i-th site; they obey the bosonic commutation relations:

[ai, a†j] =δij, [ai, aj] =0= [a†i, a†j]. (2.2)

ni is the number operator, which counts the number of bosons in the

i-th site; ei is the energy due to the external harmonic confinement to which

the bosons are undergone. The J term is the one responsible for the hopping between adjacent sites due to the tunnel coupling; it is the matrix element:

J = −

Z

d3x w(~x− ~xi)(−

∇2

2m +V(~x))w(~x− ~xj), (2.3) where w(~x− ~xi)is a single particle Wannier function localized to the i-th site,

V(~x) is the lattice potential and m is the mass of a single boson. U is instead the matrix element for the interaction between bosons in the same site; it is:

U = 4πa

m

Z

d3x |w(~x)|4, (2.4)

a being the scattering length.

If the tunnel coupling term J dominates the hamiltonian (J U), the ground-state energy is minimized if the single-particle wavefunctions of the N bosons is spread out over the entire M sites of the lattice; so the many-body ground state reads:

|ΨSFi ∝

M

∑

i=1

a†_iN|0i, (2.5)

i.e. all bosons occupy the same extended Bloch state. As pointed out by [10], an important feature of this state is that the probability distribution of the occupation ni of bosons on a single site is Poissonian, so that its variance is

given by hnii; this means that it is well described by a macroscopic

wave-function with long-range phase coherence throughout the lattice.

If the on-site interaction term U dominates (U J), the ground state of the system will instead consist of localized wavefunctions with a fixed number of bosons per site that minimized the energy; the many-body ground state will be a product of local Fock states for each site:

|ΨMIi ∝ M

∏

i=1 a†_iN|0i. (2.6)

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2.2. Quantum Simulators: Controllable Many-Body Systems 25

FIGURE2.1: Optical lattices are crystals made of light: lasers beams interfere to form

a periodic potential in which ultracold atoms are trapped. Figure taken from from the LENS laboratory webpage, Florence [13].

The two kinds of ground state have been labeled with the subscripts "SF" and "MI", specifying the two phases to which the system may be found: su-perfluid and Mott insulator phases.

2.2 Quantum Simulators: Controllable

Many-Body Systems

In order to overcome the difficulty in studying open many-body systems, over the years several analytical and numerical methods have been devel-oped. In addition to these more "traditional" approaches, during the last twenty years it arose the idea of studying a quantum system by using a properly designed quantum device. The essential aim of this new strategy lies in the employment of controlled quantum devices called quantum simula-tors; they are systems able to experimentally emulate the Hamiltonian model bearing the non-trivial properties of the system under study. The usefulness of this approach is twofold [11]: not only it allows to explore properties of the model in regions that are elusive to the analytical and numerical studies, but also it consents to test to which limits the Hamiltonian model is appropriate to describe the system or whether additional ingredients are required.

Since the Josephson junction arrays [12], that were probably the first quan-tum simulators in the history of modern physics, much progress has been made especially with the advent of cold atoms in optical lattices.

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26 Chapter 2. Quantum Simulators of Open Many-Body Systems

2.2.1 Cold Atoms in Optical Lattices

The employment of cold atoms in optical lattices has proved to be a success-ful way to create simulators of a large variety of strongly interacting sys-tems [14].

In short, optical lattices are created by making laser beams counter-propa-gating interfere with each other [15]. The result is a standing wave with a periodic pattern (see sketch in fig. 2.1). Then, ultracold atoms can be trapped in the potential wells, creating first a Bose-Einstein condensate or a cold gas of fermionic atoms and then slowly ramping up the laser beams to create the periodic lattice potential.

Ultracold atoms in optical lattices represent nearly perfect realization of fermionic and bosonic Hubbard models1[14] and they are usually employed to simulate non-dissipative systems. However, new efforts have been made in order to introduce dissipation in these devices. In fig. 2.2, a sketch of the implementation of the effective dissipative process in an optical superlattice is shown. The so-called superlattice is made up coupling driven-dissipative ultracold atoms in an optical lattice with a coherently driven reservoir.

FIGURE2.2: Schematic realization of the effective dissipative process with ultracold

atoms in an optical superlattice. Figure taken from [16].

1_{The fermionic Hubbard model in one dimension is characterized by the following}

Hamiltonian: H= −t

∑

hiji c†_iσcjσ+U

∑

i ni↑ni↓,

where ciand c†_i are the fermionic creation and annihilation operators respectively, on the i-th

site; they obey the fermionic commutation relations:

{ci, c†j} =δij, {ci, cj} =0= {c†i, c†j}.

The first term in H represents the hopping between adjacent sites and the second represents the on-site interaction between particles. The niσ are the occupation numbers, that count

the number of electron with spin σ in site i. This model is very similar to that described in section 2.1, with the difference of nature of particles considered: here they are fermions, in Bose-Hubbard model are bosons instead. In the text, the plural Hubbard models refers to the several variations of this model.

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2.2. Quantum Simulators: Controllable Many-Body Systems 27 An interesting study over the possibility of using cold atoms to simulate dissipative many-body systems, has been done by [17]. The authors of this work discuss the example of a driven dissipative Bose-Einstein condensate of bosons, where atoms in optical lattice are coupled to a bath of Bogoliubov ex-citations; the atomic current represents local dissipation. It was assumed that the atoms can be described by a Bose-Hubbard model, having the following Hamiltonian: H =H0+V ≡ −J

∑

hi,ji a†_iaj+1 2U

∑

_i a †2 i a2j, (2.7)

where H0represents the kinetic energy of bosons hopping between adjacent

lattice sites with amplitude J, V is the onsite interaction with strength U and ai (ai†) are bosonic destruction (creation) operators for atoms at i-th site. At

this point, an appropriate choice of jump operators allows to couple the sys-tem to a bath so that it is driven to a pure many-body state by quasi-local dissipation. Applying standard linearization schemes in the weakly inter-acting situations led to obtain the solution of the master equation 1.54. The steady-state solution seems to have properties similar to those of bosons in thermal contact to a heat bath.

Another strategy involves trapping Rydberg atoms in optical lattices. Rydberg atoms are alkali-like atoms with a single electron promoted into a highly excited orbital [18]. The presence of these atoms enhances photon-photon interaction; the presence of a single photon-photon in the system is able to drive away from resonance condition all atoms contained in a surrounding volume. If the optical density is sufficiently high, a second photon travelling across this volume can be absorbed or suffer a phase shift; this can be use to obtain an effective blockade effect.

2.2.2 QED-Cavity Arrays

In the last few decades the field of quantum simulators has been enriched by a new idea, based on arrays of QED cavities [11] (sketched in fig. 2.3), in which light resonates and interacts with matter contained therein. A com-pelling aspect of this kind of devices is the competition between two phe-nomena: while on the one hand light-matter interaction inside the cavity leads to photon blockade, on the other hand photon hopping between neigh-bouring cavities favors delocalization.

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FIGURE 2.3: Sketch of array of QED cavities. In the inset there is a draft of the

interaction between the two-level system and the photon resonating in the cavity and then decaying. Photons in the cavities have finite life-time, so there is a coherent

external drive.

QED cavities are mathematically described by the Jaynes-Cummings mo-del [19], in which one mode of the cavity interacts with a two-level system. By ignoring light-matter interaction, a single cavity confines several modes of electromagnetic field and each of them is quantized as a harmonic oscil-lator. If one considers a cavity with a single mode with frequency ω, the Hamiltonian describing such a system can be written as

H=ωa†_lal, (2.8)

where al(a†_l) annihilates (creates) a quantum of light in the mode of the l-th

cavity. An array of cavities sufficiently close to allow for photon hopping, can be described by eq. 2.8 to which the following term should be added:

Hhop = −J(a†lal+1+h.c.), (2.9)

where J is associated with the tunneling rate. Now, if the light-matter inter-action is introduced, the Hamiltonian can be written in this way:

H =

∑

l

H_l(0)−J

∑

hl,l0_i

(a†_lal0+h.c.), (2.10)

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2.2. Quantum Simulators: Controllable Many-Body Systems 29

FIGURE 2.4: Sketch of an array of five cavities arranged in a ring geometry with

periodic boundary conditions; the system is driven by a laser. Figure taken from [18].

we want to study QED cavities so the Hamiltonian term H_l(0) is the Jaynes-Cummings model:

H_l,JC(0) =εσ_lz+ωa†_lal+g(σ_l+al+σ_l−a†_l), (2.11) where σ_l± are the raising/lowering operators for the two-level system and ε indicates the transition energy between the two levels. The spectrum of the Hamiltonian 2.11 is anharmonic, meaning the presence of the two-level sys-tem induces a repulsion between the photons in the cavity, so that in the cav-ity only a photon can be present at the same time. Qualitatively, this can be explained by the fact that a single photon in the cavity modifies the effective resonance frequency, so that the injection of another photon is inhibited. This nonlinear process is known as photon blockade and has been named in analogy with the Coulomb blockade effect of electron transport through mesoscopic devices.

An array of cavities in the impenetrable photon regime has been theoreti-cally studied [18, 20]. As sketched in fig. 2.4, a continuous pumping is consid-ered. The transmission and/or absorption spectrum of the system shows the Tonks-Girardeau nature of the steady-state of photon gas, i.e. the fermion-ization of photons.

2.2.3 Superconducting Circuits

As for QED-cavities, also for superconducting circuits, or QED-circuits, light-matter interaction is central. In these simulators [21], the quantum "particles" are excitations, not physical particles subjected to conservation laws, so they naturally access non-equilibrium physics. Photonic modes can be realized

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FIGURE2.5: Example of an array of resonators forming a honeycomb pattern. On the

left, more than two hundred microwave cavities are coupled in a Kagome lattice, a natural two-dimensional lattice. Each cavity is coupled to two neighbours, through a capacitor that enables photon hopping. On the bottom-right, it can be seen the simmetric three-way capacitor that ensures uniform hopping rates throughout the

array. Figure taken from [21].

with on-chip microwave resonators, typically Fabry-Perot kind of cavities. A qubit coupled to such a resonator can be described by the Jaynes-Cummings model:

HJC =ωra†a+εσ+σ−+g(aσ++a†σ−), (2.12) where ωr and ε are the the photon and qubit excitation frequencies, and a†,

a, σ+ and σ− denote the corresponding raising and lowering operators. An example of these devices can be seen in fig. 2.5.

2.2.4 Arrays of Optomechanical Systems

A recent proposal by [22] considers making arrays of optomechanical sys-tems. The essential idea is that at each site of such an array, a localized mechanical mode interacts with an optical mode of a cavity driven by laser beams; the interaction occurs via radiation pressure. This way, both photons and phonons can hop between neighbouring sites. Because of the experimen-tal complexity of this kind of device, only a single cavity has been cooled to the ground state [23], so far.

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2.3. Quantum Simulation of Spin Systems 31 The work of [22] focuses on the transition of the collective mechanical motion from an incoherent state (due to quantum noise) to an ordered state with phase-coherent mechanical oscillations; for this dynamics, the dissipa-tive effects induced by the optical modes play a crucial role. Indeed, they allow the mechanical modes to have self-induced oscillations, once the op-tomechanical amplification rate exceeds the intrinsic mechanical damping; on the other hand, the quantum noise prevents the mechanical modes from synchronizing.

2.3 Quantum Simulation of Spin Systems

Interacting two level systems, either spins or qubits, are of central importance in Quantum Information and Condensed Matter Physics.

Since the discover of the photon blockade [24], it has been realized that the cavity mode is well described by a spin-1/2 Hamiltonian. Few years later, the Jaynes-Cummings model in the Mott regime was shown to simulate a XY spin model [25]. In particular, the studied system consisted of coupled electromagnetic cavities doped with a single two-level system; it has been shown that it is possible to observe an insulator phase of total (atomic plus photonic, i.e. polaritonic) excitations. From such a system, an effective XY Hamiltonian arises, with spin up (down) corresponding to the presence (ab-sence) of a polariton.

A scheme for realizing the Ising spin-spin interaction has been proposed in [26], with a system consisting in trapped two-level atoms in one dimensio-nal coupled microcavities.

A more complicated system has been investigated in [27], which uses three-level atoms in micro-cavities coupled to each other via the exchange of virtual photons; it is shown that it can model an anisotropic Heisenberg spin-1/2 chain in an external magnetic field. The two spin polarizations|↑i

and|↓iare represented by two long-lived atomic levels of aΛ level-structure; let us see why. The quantized mode of the cavity together with external lasers can induce a Raman transition between these two lowest-lying states. With appropriately chosen detunings, the dominant Raman transition between the two lowest-lying states involves one laser and one cavity photon. This im-plies that the emission and absorption of virtual photons in the cavity is con-fined to only two states per atom, the long-lived ones, so that it can be de-scribed by a spin-1/2 Hamiltonian. The coupling of virtual photons between neighbouring cavities simulates σxσxand σyσyinteractions; the effective spin

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32 Chapter 2. Quantum Simulators of Open Many-Body Systems can be coupled to an effective magnetic field σz; using the same atomic level configuration but a different set-up of the external sources an effective σzσz interaction is obtained.

FIGURE2.6: Simplified schematic of a superconducting flux qubit

acting as a quantum mechanical spin [28].

By means of superconducting circuits, it has been possible to simulate small spin chains, by directly coupling a number of superconducting qu-bits [28]. Fig.2.6 shows two superconducting loops in the qubit, each sub-ject to an external flux biasΦ1x and Φ2x. The dynamics of the device can be

modelled as a quantum mechanical double-well potential. The two lowest energy states of the system correspond to clockwise or anticlockwise circu-lating current in loop 1. Considering only these two states (in the limit of low temperature it is a valid approximation), the qubit dynamics is that of an Ising spin. Qubits are then coupled together using programmable coupling elements; this allows to tune the spin coupling in a continous way between ferromagnetic and antiferromagnetic character. The behaviour of this system is well described by an Ising model Hamiltonian:

H = N

∑

i=1 hiσiz+ N

∑

i,j=1 Jijσizσjz. (2.13)

In this way, it has been possible to simulate a one-dimensional 8-sites spin chain, with the ends polarized in opposite directions.

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33

Chapter 3

Numerical Methods for Open

Quantum Systems

Hilbert space is a big place.

Carlton Caves

As we have seen in chapter 1, the study of an open quantum system re-quires the knowledge of its density matrix, solution of the Lindblad master equation [5]: ˙ρ= L[ρ], where L[ρ] ≡ −i[H, ρ] − N2−1

∑

a=1 γa 1 2L † aLaρ+1 2ρL † aLa−LaρL†_a , (3.1)

N being the dimension of the Hilbert space.

In such a system the number of variables scales as the square of the di-mension of the Hilbert space; for example, in a spin-1/2 system made up by n elements, the size of the Hilbert space H is 2n while the liouvillian di-mension equals to 22n ×22n. Clearly, a direct numerical integration of the equation 3.1 can be made only for systems with a very limited number of elements. One only needs to note the fact that for a brute-force integration of the equation 3.1 for a system consisting of 8 sites, about 34 GB of memory would be necessary1.

Nevertheless, some techniques for exact diagonalization of the Liouvil-lian superoperator 3.1 have been developed. Let us mention, for example, the one devised by Prosen [29], for studying a system characterized by a

1_{This calculation considers the fact that the liouvillian contains, in general, complex}

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34 Chapter 3. Numerical Methods for Open Quantum Systems quadratic Hamiltonian and linear Lindblad operators, which prescribes to di-agonalize the Liouville super-operator in terms of 2n normal master modes, i.e. anticommuting super-operators which act on the Fock space of density operators. This can be understood as a complex version of the Bogoliubov transformation.

However, as seen previously, a direct integration of the master equation for an open many-body system becomes unfeasible even for relatively small sizes, so conceiving more clever numerical strategies seems to be necessary.

One of them is based on the mean field approach; the essential idea is to consider the mean value of the Hamiltonian and, as suggested by Gutzwiller, to factorize the density matrix, defining a mean field density matrix ρMF =

∏jρj, where ρj is the single site density matrix. In this way, all correlations

and interactions between sites are neglected, so we can say it is a crude ap-proximation for an interacting many-body system. In order to (partially) overcome this problem, the cluster mean field method was developed: in this method a set of contiguous sites are isolated from the rest of the system. As explained in [30], this method suggests to write the Hamiltonian as HCMF =

HC +HB(C), where the first term describes the exact interactions inside the cluster C while the second term represents the mean-field interactions be-tween the cluster and the sites on its boundary B(C). The global density matrix will be ρCMF = NρC, where the ρC is the density matrix of the C-th cluster. One should note that, when the cluster is made up by a single site, the method becomes the above-mentioned standard one.

Other important numerical methods are the ones essentially based on the linked cluster expansions [31], in which the quantities of interest are expressed as sums of contributions from a sequence of clusters of sites; in particular, only connected clusters contribute. In [32] an application of this idea to the Liouvillian is presented.

In this section, we will examine some of the numerical techniques used for studying open quantum systems with very different approaches.

Two of the three numerical methods analyzed in this chapter are essen-tially based on the Wilson’s renormalization group approach [33] combined with the fundamental contribution of the White’s density matrix renormaliza-tion group (DMRG) method [34] and are the corner-space renormalizarenormaliza-tion and the matrix product density operators methods. They both use the idea of a sys-tem decomposition into blocks; even if the basic idea is the same, the devel-oped approaches are actually different.

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3.1. The Quantum Trajectories (QT) Method 35 The first numerical method analyzed in the section 3.1 is the quantum tra-jectories method, based on the evolution of a Monte Carlo wave function.

3.1 The Quantum Trajectories (QT) Method

The quantum trajectories method is an estabilished numerical method, in which pure states are the subjects of the study, instead of density matrices. This means that, if the Hilbert space has dimension N, the number of involved parameters (∼ N) is much smaller than the one required in calculations with density matrices (∼ N2). This method was originally proposed in 1992 by Dalibard, Castin and Mølmer in [35], then generalized in [36], as a stochas-tic unraveling of the master equation, namely a stochasstochas-tic evolution for the wave function of a system coupled to a reservoir. It has been proved [35, 36] that this Monte Carlo wave-function approach is equivalent to the master equation treatment.

The idea of this method can be summarised in the following way.

First of all, given the 3.1, let us note that the first three terms of this equation can be regarded as the evolution performed by an effective non-Hermitian Hamiltonian, that is [37]:

Heff =Hs+iK, (3.2) with K = −1 2

∑

_µ L † µLµ. (3.3)

Indeed, we see that:

−i[Heff, ρ] = −i[Hs, ρ] −1

2

∑

_µ {L

†

µLµ, ρ}.

The summation term in the 3.1, is the one responsible for the so-called quantum jumps; for this reason, the representation under which we have writ-ten the 3.1 is called quantum jump picture [37].

At an initial time t0, the density matrix of the system is in a pure state ρ(t0) = |φ(t0)i hφ(t0)|;

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36 Chapter 3. Numerical Methods for Open Quantum Systems after a time dt, it evolves to the following statistical mixture:

ρ(t0+dt) = 1−

∑

µ dpµ |φ0i hφ0| +

∑

µ dpµ|φµi hφµ|, (3.4) where dpµ = hφ(t0)|Lµ†Lµ|φ(t0)idt (3.5)

is the probability that a jump occurs; in this case, the system evolves into the state

|φµi =

Lµ

Lµ|φ(t0)i

|φ(t0)i. (3.6)

Otherwise, if no jump happens, the system evolves according to the effective Hamiltonian Heffin the following way:

|φ0i =

(1−iHeffdt)

q

1−_∑_µdpµ

|φ(t0)i. (3.7)

In order to decide if the jump occurs or not, a Monte Carlo method will be integrated in this picture. Namely, an uniform distribution in the unit interval [0, 1]is taken under consideration; for every experiment, a pseudo-random number e is chosen from this uniform distribution, i.e. a coin is tossed: depending on the result of the throw, the possible situations are the following:

• if e<∑_µdpµ, the system jumps to one of the states|φµi, defined in 3.6.

In particular:

– if 0≤e ≤dp1, the system jumps to|φ1i;

– if dp1<e ≤dp2, the system jumps to|φ2i;

– and so on;

• if e>_∑_µdpµ, the system evolves to the state|φ0i.

This process has to be repeated as many times as n = _dtT, where T is the whole elapsed time during the evolution. Let us note that dt must be taken much smaller than the scales relevant for the evolution of the system.

Every experiment, i.e. every throw of the coin, gives a different quantum trajectory, which can be used to calculate the mean value of an observable A at a certain time t, in this way:

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3.2. The Corner-Space Renormalization (CSR) Method 37 Since this results from a Monte Carlo process, we can consider the mean value over N experiments:

hA(t)i = lim N→∞ 1 N N

∑

i=1 hφi(t)|A|φi(t)i. (3.9)

So, to sum up we can say a few things about QT algorithm; it allows to do the so-called stochastic unraveling of the master equation, using pure states instead of density matrices: as we have seen, this simplifies the computa-tional complexity of the problem. However, this method has an important limit: it does not prevent the exponential growth with respect to the size of the system. The methods analyzed in the next two sections are specifically constructed to face with this issue.

3.2 The Corner-Space Renormalization (CSR)

Method

The numerical methods based on renormalization group à la Wilson are char-acterized by the fact that the dimension of the Hilbert space increases, while two blocks are merged; the fundamental aim of the corner-space renormal-ization (CSR) method [38] is to deal with this problem and try to overcome the limitation mentioned above.

FIGURE3.1: Blocking scheme of Wilson’s NRG method.

First of all, it is important to mention Wilson’s numerical renormalization group (NRG) procedure, in which the CSR method is rooted.

Let us consider a 1D-system made up by N spins. The NRG method sug-gests to take two spins and consider a subspace of the corresponding Hilbert spaceH2 ⊂ H1⊗ H1 of dimension d2 ≤ d2₁; then, one adds a spin and

con-sider the subset of the Hilbert space of three spinsH3 ⊂ H2⊗ H1 of

dimen-sion d3 ≤ d2d1, and so on until one arrives to consider the subset of Hilbert

spaceH_N ⊂ H_N−1⊗ H1with dimension dN ≤ dN−1d1. So, one can

approxi-mate the Hamiltonian with PNHPN, where the PNis the projector ontoHN. If

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38 Chapter 3. Numerical Methods for Open Quantum Systems find eigenvalues and eigenvectors. A good thing to do is fixing for all the dimensions dn the so-called bond dimension such that

dn ≤D ∀ n.

Considering a one-dimensional spin chain, the NRG method requires to break the chain into a set of identical blocks A (see fig. 3.1). So, one diagonal-izes the Hamiltonian matrix HAA for two neighboring blocks A and uses its

lowest m eigenstates (ordered by energy) to form a pseudo-unitary matrix O, employed to change the basis and make up the Hamiltonian HA0

represent-ing the block A0twice as large (see fig. 3.1), in this way: HA0 =OH_AAO†,

where O is a m×n matrix (the rows of O are the m lowest eigenstates of HAA)

and HAA has dimension n×n. This procedure has to be repeated using the

new larger blocks A0and the new effective Hamiltonian HA0.

FIGURE3.2: Merging of the blocks in NRG procedure.

It is worth noting that the NRG method overlooks the connections be-tween the blocks (see fig. 3.2), so it is a good method only in the limit that the links between the blocks vanish. This weak point has been studied and overcome by White in [34] where he developed the DMRG method, that will be outlined in the following section.

The CSR method can be seen as a generalization of the NRG method, a development to the case of open systems. The name refers to the idea of selecting a corner of the Hilbert space for a lattice system, using eigenvectors of the steady-state density matrix of smaller lattices.

Let us start considering two blocks of a quantum system consisting in a certain number N of sites. The CSR approach is a recursive process that begins with the calculation of the steady-state density matrix of a small block

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3.2. The Corner-Space Renormalization (CSR) Method 39 (the first block will be a single site, in the second iteration it will be a small lattice constituted of the two blocks emerged in the previous iteration, and so on); this calculation can be done studying the spectrum of the eigenvalue of the Liouvillian and taking the zero value, considered that

dρ

dt = Lρ (3.10)

can be seen as an eigenvalue equation.

We now consider two spatially adjacent sites: the site A and the site B. We now call ρA the steady-state density matrix of the first site and ρB of the

lat-ter site. In order to spatially merge these two sites, let us write ρAand ρB in

terms of an orthonormal basis; if there is no degeneracy among eigevalues of ρA and ρB, we can use the orthonormal basis formed by their eigenvectors.

Instead, if there is degeneration, a Gram-Schmidt orthonormalization pro-cess can be used to obtain an orthonormal basis starting from the ensemble of eigenvectors.

In this way, we will have:

ρA =

∑

r pr(A)|φ (A) r i hφ (A) r |, ρB =

∑

r0 p_r(0B)|φ (B) r0 i hφ (B) r0 |, (3.11) where p(rA) ≥0 ∀r and p (B) r0 ≥0 ∀r0,

∑

r p(rA) =1 and

∑

r0 p(_rB0 ) =1.

The steady-state density matrix of the merged block constituted of the block A and the block B will be the following:

ρAUB =ρA⊗ρB =

∑

r

∑

r0 pr(A)p (B) r0 |φ (A) r i |φ_r(0B)i ⊗ hφ (B) r0 | hφ (A) r | (3.12)

At this point, the CSR method suggests to delineate the corner-space; for this purpose, we consider the M most probable product states coming from equation 3.12 of the form |φ(rA)i |φ_r(0B)i, i.e. we rank them according to the

joint probability p(rA)p

(B)

r0 . The M states form an orthonormal basis that

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40 Chapter 3. Numerical Methods for Open Quantum Systems

FIGURE3.3: Sketch of the corner-space renormalization method.

called corner-space, with pr(₁A)p(_r0B) 1 ≥ pr₂(A)p(_rB0 ) 2 ≥ · · · ≥ pr(_MA)p(_r0B) M.

So, now it becomes necessary doing a change of basis to the new block A∪B, that is:

H0 =OHe _A_∪_BOe†, (3.14)

where eO is a pseudo-unitary2 M×N matrix, N is the dimension of HA∪B;

e

O is formed by the M eigenvectors of ρA∪B living in the corner-space. This

density matrix can now be used to calculate the expectation values of any observable.

The larger is M, the closer to exactness are the results, because the ba-sis 3.13 spans a larger part of the Hilbert space. Therefore, at this point the value of M should be increased until the convergence is reached.

In the work of [38] it has been proved that convergence can be achieved with a number of states M much smaller than the dimension of the Hilbert space. In the present thesis, it will be shown that this method can not be used for the analysis of systems described by a model such as that under consideration in the present work.

In short, the algorithm is structured in the following stages, as sketched in fig. 3.3:

calculation of the steady-state density matrix of a single block of the system;

Nonequilibrium Steady-State Analysis of a Boundary-Driven Interacting Quantum Spin Chain

T

ESI DI

L

AUREA

M

AGISTRALE

Nonequilibrium Steady-State

Analysis of a Boundary-Driven

Interacting Quantum Spin Chain

Candidata:

Francesca C

Relatore:

Dr. Davide R

Contents

Introduction

Chapter 1

Open Quantum Systems

1.1

Mathematical Background

1.1.1

Pure and Mixed States: the Density Matrix Formalism

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

1.1.2

Closed Quantum Systems

∑

∑

1.1.3

Open Quantum Systems

∑

1.2

Approximate Description of

Open Quantum Systems

1.2.1

Quantum Dynamical Semigroups

∑

∑

∑

∑

∑

1.2.2

The Markovian Evolution

1.2.3

The Lindblad Master Equation

∑

∑

∑

∑

∑

∑

1.3

Quantifying Correlations in Quantum Systems

1.3.1

Bipartite Pure States

∑

∑

1.3.2

Bipartite Mixed States

∑

∑

∑

Chapter 2